Is the following true for exponential independent random variables $X_i$ with CDF $\mathbb{P}\left( X_i \leq x\right)=1-\frac{1}{\sqrt{t}}e^{- \frac{x}{\sqrt{t}}}$? $$ \lim_{t \to \infty} \mathbb{P}\left(\sum_{i=1}^{\sqrt{t}/2} X_i \leq t\right)=1$$
Since $\mathbb{E}[X_i]=\sqrt{t}$ that goes to infinity as $t \to \infty$, we cannot use the weak law of large numbers, central limit theorem, or the Chebyshev inequality. Any other idea?